Finding y-intercept

Question

I am having trouble finding the solution to e^x-3e^-x-4x=0. I know the answer is roughly 2.2 but am not sure how to get there?

If anyone could help me in the right direction with working that would be greatly appreciated !

Answer 1

Outside of a few lucky cases, it is not possible to exactly solve equations that mix exponential and non-exponential terms. However, you can use a variety of methods to find increasingly good approximations.

If you don't know any calculus, then you can still apply the bisection method:

1. Let $f(x) = e^x + 3e^{-x} - 4x$

2. Find $a, b$ such that $a < b$ and either $f(a) < 0 < f(b)$ or $f(a) > 0 > f(b)$ (i.e. find two points such that the function changes sign between them, meaning there is a solution somewhere in that interval - this only works because $f$ is a continuous function).

3. Take $c = \frac{1}{2}(a + b)$. If $f(c) = 0$ then $c$ is the desired zero, and you can stop. Or, if $f(c)$ is smaller than the error you're willing to accept, then it's your approximation and you can stop.

4. Find the sign of $f(c)$. If $f(a)$ and $f(c)$ have the same sign, set $a := c$, otherwise set $b := c$ (i.e. pick a new $a$ and $b$ such that you still know that the zero is between them).

5. Go back to step 2 with your new $a$ and $b$.

If you know a little calculus, you can apply a different method like Newton-Raphson. Or you can check out the linked Wikipedia article for a few other alternatives.